Global pinching theorems for minimal submanifolds in spheres

Kairen Cai

Kairen Cai

Colloquium Mathematicae, Tome 96 (2003), p. 225-234 / Harvested from The Polish Digital Mathematics Library

Résumé

Let M be a compact submanifold with parallel mean curvature vector embedded in the unit sphere $S^{n + p} (1)$ . By using the Sobolev inequalities of P. Li to get $L_{p}$ estimates for the norms of certain tensors related to the second fundamental form of M, we prove some rigidity theorems. Denote by H and ${| | σ | |}_{p}$ the mean curvature and the $L_{p}$ norm of the square length of the second fundamental form of M. We show that there is a constant C such that if ${| | σ | |}_{n / 2} < C$ , then M is a minimal submanifold in the sphere $S^{n + p - 1} (1 + H ²)$ with sectional curvature 1+H².

Publié le : 2003-01-01

Zbl 1046.53035

EUDML-ID : urn:eudml:doc:284521

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     title = {Global pinching theorems for minimal submanifolds in spheres},
     journal = {Colloquium Mathematicae},
     volume = {96},
     year = {2003},
     pages = {225-234},
     zbl = {1046.53035},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm96-2-7}
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Kairen Cai. Global pinching theorems for minimal submanifolds in spheres. Colloquium Mathematicae, Tome 96 (2003) pp. 225-234. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm96-2-7/